# Chapter 2.5  Continuity at a Point: A function is called continuous at c if the following 3 conditions are met.  1) is defined 2) exists 3)  Continuity.

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Chapter 2.5

 Continuity at a Point: A function is called continuous at c if the following 3 conditions are met.  1) is defined 2) exists 3)  Continuity on an Open Interval: A function is called continuous on an open interval (a,b) if it is continuous at each point in the interval. A function that is continuous on the entire real line is called everywhere continuous Definition of Continuity

 A function is said to be discontinuous at c if is defined on an open interval containing c (except possibly at c) and f is not continuous at c.

 1) Removable: A discontinuity at x=c is removable if can be made continuous by appropriately defining (or redefining) at x=c.  2) Nonremovable: For example, if the graph is broken or has gaps

 If is continuous on the closed interval [a,b] and k is any number between and then there is at least one number c in (a,b) such that.  Note: If is continuous on [a,b] and and differ in sign, then the IVT guarantees the existence of at least one zero of in the closed interval [a,b].

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